Neighbor Distribution Function Research Articles

Nearest neighbor spatial statistics as a framework for analyzing spatial patterns from image data.

Quantitatively describing spatial patterns in biological substrates is a difficult task that requires careful consideration of labeling methods, imaging modality, and the motivating hypothesis. This problem is commonly addressed using intensity correlation paradigms which correlate the brightness between fluorescence images of co-labeled biomolecules as a measure of their colocalization. These approaches ultimately characterize the superposition of fluorescence signals and thus are limited to binary assessment of spatial relationships at the scale of the image resolution. We’ve previously demonstrated an image analysis framework, which leverages nearest neighbor (NN) distribution functions (F, G, and L) for richer, more informative quantification of the spatial distribution of immunofluorescently imaged biomolecules on all spatial scales captured within the image. Briefly, biomolecule signals and relevant biological structures are segmented, biomolecules are localized relative to landmarks, which are either structures or other biomolecules, by measuring NN distances between biomolecule and landmark voxels, and finally their global spatial association is assessed by comparing the resulting distance distribution to a null hypothesis distribution, in our case a homogenous Poisson distribution which represents complete spatial randomness. The sign of the difference between the distributions indicates repulsion or attraction of the biomolecule to the landmark and the magnitude indicates the relationship's strength. Ongoing efforts aim to further validate this framework by characterizing its sensitivity to image segmentation parameters and signal concentration, and the impact on measured spatial relationships of our methods for generating the null distribution and measuring distances within an image of anisotropic resolution. Preliminary results indicate our framework is robust over a wide range of input parameters and data quality, and our custom methods introduce errors that are smaller than the image's precision. Thus, this approach has wide applicability for scientific image analysis.

Structure of cosmic web in non-linear regime: the nearest neighbour and spherical contact distributions

ABSTRACT In non-linear scales, the matter density distribution is not Gaussian. Consequently, the widely used two-point correlation function is not adequate anymore to capture the matter density field’s entire behaviour. Among all statistics beyond correlation functions, the spherical contact (or equivalently void function), and nearest neighbour distribution function seem promising tools to probe matter distribution in non-linear regime. In this work, we use haloes from cosmological N-body simulations, galaxy groups from the volume-limited galaxy group and central galaxies from mock galaxy catalogues, to compare the spherical contact with the nearest neighbour distribution functions. We also calculate the J-function (or equivalently the first conditional correlation function), for different samples. Moreover, we consider the redshift evolution and mass-scale dependence of statistics in the simulations and dependence on the magnitude of volume-limited samples in group catalogues as well as the mock central galaxies. The shape of the spherical contact probability distribution function is nearly skew-normal, with skewness and kurtosis being approximately 0.5 and 3, respectively. On the other hand, the nearest neighbour probability distribution function is nearly lognormal, with logarithmic skewness and kurtosis being approximately 0.1 and 2.5, respectively. Accordingly, the spherical contact distribution function probes larger scales compared to the nearest neighbour distribution function, which is influenced by details of structures. We also find a linear relation between the mean and variance of the spherical contact probability distribution function in simulations and mock galaxies, which could be used as a distinguishing probe of cosmological models.

The impact of coppice management on spatial structure and intraspecific interactions of Brant's oak (Quercus brantii Lindl.) semi-arid woodlands

Coppicing is a common management strategy in oak woodlands; however, our knowledge is limited in the impacts of coppicing on spatial structure and growth dynamics of oak stands. In this study, we compared the spatial patterns of two adjacent high and coppice pure stands of Brant's oak (Quercus brantii Lindl.) in central part of Zagros semi-arid woodlands, western Iran. We used univariate form of scale-dependent summary statistics (nearest neighbour distribution function, L-function, pair- and mark correlation functions) and Analytical Global Envelopes under inhomogeneous Poisson process which allow detection of the intraspecific interactions of the individuals within the high and coppice stands. Our results indicated the aggregation of oak individuals at small spatial scales (i.e., 0-3 m) in the high stand, most likely as a result of aggregation in parts with environmentally benign conditions. The positive interactions of conspecifics had a marked effect on their size (e.g., larger height (5%), DBH (24%), and crown area (50%) than mean size at spatial scale of 2 m) inferred as growth stimulation and aggregation of young trees around their nurse or parent trees. However, the findings revealed the segregation of oak trees at short distances in the coppice stand (i.e., 0–5 m), probably due to strong competition of conspecific neighbors. The negative size correlation of oaks (e.g., smaller height, DRC, and crown area of oak individuals (15%) than mean size at distance of 2 m) was considered as the outcome of competition for scarce resources in the woodlands with unfavourable conditions. In general, it was concluded that the aggregated spatial structure of the high stand has changed to segregation of individuals within the coppice stand. Additionally, we found that environmental heterogeneity and intraspecific competition are the main drivers of the high and coppice stands, respectively.

A foraging ocean sunfish and the ‘nearest neighbor’ problem

A predator that preys on randomly-distributed stationary energetically-equivalent small prey will probably choose its next prey to be the nearest one. But what if no prey is found within the detection range of the predator? It is hypothesized that in this case the predator will move along an arbitrary chosen direction until a prey is detected, and turn towards it. In a stochastic environment this strategy leads to a certain distribution function of distances that the predator moves between consequent prey catches. It is shown that when the detection range of the predator exceeds the average distance between prey, this distribution function becomes the nearest neighbor distribution function, whereas; wherew when the detection range is small as compared with the average distance between prey, it becomes the exponential distribution, as the distribution of distances between neighbors on a line. In the first case, the average distance between catches becomes roughly half the average distance between prey; in the second case, it becomes inversely proportional to the square of the detection range. Ocean sunfish preys on practically stationary jellyfish at depth of more than a hundred meters, in dim light. Plausibly, it can detect jellyfish only at close quarters, and hence its detection range is probably small as compared with the average distance between prey. Analysis of the tracking data from seven animals over a few days yielded many thousands of swimming segments separating consequent prey catches. Indeed, lengths of these segments were shown to have the exponential distribution. This finding not only supports the initial hypothesis of this study, but also reveals the fragility of the energetic balance of this animal. A two-fold decrease in the detection range (e.g. due to a decreased visibility) is expected to increase the average distance it moves between catches four-fold, and hence decrease its specific energy intake (the number of jellyfishes per distance moved) by the same rate.

Diffraction patterns of amorphous materials as a series expansion of neighbor distribution functions

An exact analytical expression for the static structure factor in disordered materials is derived from Fourier transformed neighbor distribution decompositions in real space, and permits to reconstruct the function in an iterative fashion. The result is successfully compared to experimental data of archetypal glasses or amorphous materials (GeS2, As2Se3, GeTe), and links quantitatively knowledge of structural information on short and intermediate -range order with the motifs found on the diffraction patterns in reciprocal space. The approach furthermore reveals that only a limited number of neighbor shells is sufficient to reasonably describe the structure factor for k > 2 . In the limit of the high momentum transfer, the oscillation characteristics of the interference function are related with new informations on the short-range order of disordered materials.

Nearest Neighborhood Characteristics of a Tropical Mixed Broadleaved Forest Stand

Structural complexity and local biodiversity of species-rich tropical forests can be characterized by their spatial patterns, which contribute to species intra- and interspecific interactions. Aiming to describe spatial patterns of species at fine spatial scales, we applied the quantitative analyses based on the relationships of nearest neighbors of conspecific and heterospecific trees. In a two-hectare plot of a tropical broadleaved forest stand in central Vietnam with minimal human influence, all tree individuals with diameter at breast height ≥ 2.5 cm were mapped and their characteristics were recorded. We applied two different types of analyses: (1) Intraspecific structural characteristics using nearest neighbor statistics; (2) overall interspecific associations through a classification scheme based on bivariate nearest neighbor distribution function D12(r) and Ripley’s K function K12(r). The findings showed that: (1) Most of studied species in the forest were highly mixed with other species, while conspecifics were regular to aggregated distribution at small spatial scales. Tree individuals with different diameter values were surrounded by heterospecific trees; (2) The majority of 306 species-species pairs showed spatial independence (66.7%), whereas 29.8% of all species showed an overall positive association and negative association consisted only a small percentage (3.5%) up to spatial scales of 50 m. We found significant evidences of the main ecological theories such as dispersal limitation, Neutral theory, Janzen-Connell hypothesis, and other effects like the stochastic dilution. We suggest using both the bivariate distribution of the structural parameters and the spatial point pattern analysis based on nearest neighbor distance as advantageous approaches for further understanding of population structure, as well as discovering and protecting biodiversity in the future.

Spatial distribution of nuclei in progressive nucleation: Modeling and application

Phase transformations ruled by non-simultaneous nucleation and growth do not lead to random distribution of nuclei. Since nucleation is only allowed in the untransformed portion of space, positions of nuclei are correlated. In this article an analytical approach is presented for computing pair-correlation function of nuclei in progressive nucleation. This quantity is further employed for characterizing the spatial distribution of nuclei through the nearest neighbor distribution function. The modeling is developed for nucleation in 2D space with power growth law and it is applied to describe electrochemical nucleation where correlation effects are significant. Comparison with both computer simulations and experimental data lends support to the model which gives insights into the transition from Poissonian to correlated nearest neighbor probability density.

Robust nonparametric quantification of clustering density of molecules in single-molecule localization microscopy.

We report a robust nonparametric descriptor, J′(r), for quantifying the density of clustering molecules in single-molecule localization microscopy. J′(r), based on nearest neighbor distribution functions, does not require any parameter as an input for analyzing point patterns. We show that J′(r) displays a valley shape in the presence of clusters of molecules, and the characteristics of the valley reliably report the clustering features in the data. Most importantly, the position of the J′(r) valley () depends exclusively on the density of clustering molecules (ρc). Therefore, it is ideal for direct estimation of the clustering density of molecules in single-molecule localization microscopy. As an example, this descriptor was applied to estimate the clustering density of ptsG mRNA in E. coli bacteria.

Competitive interactions of Persian oak coppice trees (Quercus brantii var. persica) in a pure dry woodland revealed through point pattern analysis

Understanding intraspecific interactions among Persian oaks is essential to predict the responses of Zagros dry woodlands purely covered by these coppice trees to environmental changes (such as drought) that have direct effects on the sustainability of these woodlands. This investigation attempted to explore the scale-dependent competition among Persian oaks and highlight the impacts of competitive interactions on their biophysical properties (i.e. tree height, diameter at root collar and crown attributes) via point pattern analysis. Structurally different summary statistics were implemented to analyse the spatial pattern of Persian oaks in a pure 9-ha study plot. Nearest neighbour distribution function D(r) indicated that these coppice trees had no nearest neighbours up to the spatial scale of 2 m, and they all had a nearest neighbour at distances shorter than 12 m. Furthermore, pair correlation function g(r) showed the significant dispersion of Persian oaks up to the scale of 5 m, which may be the outcome of intraspecific competition for resources such as soil nutrients and water. As explored by mark variograms γ m (r), Persian oaks of the same size had significant spatial correlations that illustrated the competitive (not facilitative) interactions of these coppice trees in the study plot. Mark correlation function k mm (r) additionally revealed negative correlations of height and radial growth up to large spatial scales. Consequently, the outcome of this study highlighted the significant effects of competitive intraspecific interactions among Persian oak coppice trees on their spatial pattern and biophysical properties.

Robust Nonparametric Descriptors for Clustering Quantification in Single-Molecule Localization Microscopy

Single-molecule localization microscopy (SMLM) has been utilized broadly in imaging biological molecules in various biological systems, allowing quantitative analyses on the spatial organizations and patterns of these molecules. However, parameters are needed in many of the currently available methods or algorithms, likely introducing subjective bias in the analyses. Here, we report a robust nonparametric descriptor, J'(r) , for quantifying the spatial organization of molecules in single-molecule localization microscopy. J'(r) , based on nearest neighbor distribution functions, does not require any parameter as an input for analyzing point patterns. We show that it displays a valley shape in the presence of clusters of molecules, and the characteristics of the valley reliably report the clustering features in the data. More importantly, the position of the J'(r) valley (rJ'm) depends exclusively on the density of clustering molecules. Therefore, it is ideal for direct measurements of clustering density of molecules in single-molecule localization microscopy.

Nonpolar Solvation Free Energy from Proximal Distribution Functions.

Using precomputed near neighbor or proximal distribution functions (pDFs) that approximate solvent density about atoms in a chemically bonded context one can estimate the solvation structures around complex solutes and the corresponding solute-solvent energetics. In this contribution, we extend this technique to calculate the solvation free energies (ΔG) of a variety of solutes. In particular we use pDFs computed for small peptide molecules to estimate ΔG for larger peptide systems. We separately compute the non polar (ΔGvdW) and electrostatic (ΔGelec) components of the underlying potential model. Here we show how the former can be estimated by thermodynamic integration using pDF-reconstructed solute-solvent interaction energy. The electrostatic component can be approximated with Linear Response theory as half of the electrostatic solute-solvent interaction energy. We test the method by calculating the solvation free energies of butane, propanol, polyalanine, and polyglycine and by comparing with traditional free energy simulations. Results indicate that the pDF-reconstruction algorithm approximately reproduces ΔGvdW calculated by benchmark free energy simulations to within ∼ kcal/mol accuracy. The use of transferable pDFs for each solute atom allows for a rapid estimation of ΔG for arbitrary molecular systems.

MODELING THE SPATIAL DISTRIBUTION OF ESHNAN (SEIDLITZIA ROSMARINUS) SHRUBS TO EXPLORING THEIR ECOLOGICAL INTERACTIONS IN DRYLANDS OF CENTRAL IRAN

Abstract. Evaluating the interactions of woody plants has been a major research topic of ecological investigations in arid ecosystems. Plant-plant interactions can shift from positive (facilitation) to negative (competition) depending on levels of environmental stress and determine the spatial pattern of plants. The spatial distribution analysis of plants via different summary statistics can reveal the interactions of plants and how they influence one another. An aggregated distribution indicates facilitative interactions among plants, while dispersion of species reflects their competition for scarce resources. This study was aimed to explore the intraspecific interactions of eshnan (Seidlitzia rosmarinus) shrubs in arid lands, central Iran, using different summary statistics (i.e., pair correlation function g(r), O-ring function O(r), nearest neighbour distribution function D(r), spherical contact distribution function Hs(r)). The observed pattern of shrubs showed significant spatial heterogeneity as compared to inhomogeneous Poisson process (α=0.05). The results of g(r) and O(r) revealed the significant aggregation of eshnan shrubs up to scale of 3 m (α=0.05). The results of D(r) and Hs(r) also showed that maximum distance to nearest shrub was 6 m and the distribution of the sizes of gaps was significantly different from random distribution up to this spatial scale. In general, it was concluded that there were positive interactions between eshnan shrubs at small scales and they were aggregated due to their intraspecific facilitation effects in the study area.

Spatial patterns of primary seed dispersal and adult tree distributions: Genipa americana dispersed by Cebus capucinus – CORRIGENDUM

Within the second paragraph of page 494 incorrect language was used to characterize the summary characteristics used. Sentences 3–11 of this paragraph should have read:Second, we calculated three univariate summary characteristics: the nearest neighbour distribution function D(r), the pair-correlation function g(r) and the K-function K(r). The use of multiple summary characteristics holds increased power to characterize variation in spatial patterns (Wiegand et al. 2013). The univariate nearest neighbour distribution function D(r) can be interpreted as the probability that the typical adult tree has its nearest neighbouring adult tree within radius r (or alternatively, the probability that the typical defecation has its nearest neighbouring defecation within radius r). The univariate pair-correlation function g(r) is a non-cumulative normalized neighbourhood density function that gives the expected number of points within rings of radius r and width w centred on a typical point, divided by the mean density of points λ in the study region (Wiegand et al. 2009). We applied g(r) to trees and defecation point patterns separately, using a ring width of 10 m. The K-function K(r) provides a cumulative counterpart to the non-cumulative pair-correlation function g(r) by analysing dispersion and aggregation up to distance r rather than at distance r (Weigand & Moloney 2004). The K-function can be defined as the number of expected points (i.e. either trees or defecations) within circles of radius r extending from a typical point, divided by the mean density of points λ within the study region. Here, we apply the square root transformation L(r) to the K-function to remove scale dependence and stabilize the variance: $L( r ) = \scriptstyle\sqrt {\frac{{K( r )}}{\pi }} - r$ (Besag 1977, Wiegand & Moloney 2014).

Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares.

We extend the transfer matrix method of one-dimensional hard core fluids placed between confining walls for that case where the particles can pass each other and at most two layers can form. We derive an eigenvalue equation for a quasi-one-dimensional system of hard squares confined between two parallel walls, where the pore width is between σ and 3σ (σ is the side length of the square). The exact equation of state and the nearest neighbor distribution functions show three different structures: a fluid phase with one layer, a fluid phase with two layers, and a solid-like structure where the fluid layers are strongly correlated. The structural transition between differently ordered fluids develops continuously with increasing density, i.e., no thermodynamic phase transition occurs. The high density structure of the system consists of clusters with two layers which are broken with particles staying in the middle of the pore.

An application of a spatial simulated annealing sampling optimization algorithm to support digital soil mapping

Spatial simulated annealing (SSA) was applied to optimize the sampling confi guration for soil organic matt er mapping through various sampling scenarios in a Hungarian study site. Prediction-error variance of regression kriging was applied as quality measure in the optimization procedures. Requisites of SSA come from a legacy soil dataset and from spatial auxiliary information. Four scenarios were set to represent the major capabilities of SSA. Scenario 1 and 2 represented completely new sampling designs to optimize with predefi ned constraints. In scenario 1, number of new observations was the constraint, whilst in scenario 2, it was the value of the quality measure. In both scenarios, areas inaccessible for sampling (roads, farms etc.) were also taken into account. Scenario 3 and 4 represented complementary sampling confi gurations to optimize taking the previously collected samples into consideration. In scenario 3, the constraint was the number of new observations, whilst in scenario 4, it was the value of the quality measure. In both cases, two types of previously collected sampling design were simulated, a regular and a clustered confi guration. The resulted designs were evaluated by Kolmogorov–Smirnov test, nearest neighbour distribution function and empty space function. In cases of scenario 1 and 3, the results showed that, all of the optimized sampling confi gurations cover properly both geographic and feature space, respectively. In cases of scenario 2 and 4, the resulted calibration curves can be used to determine the sample size for a given quality measure value. Furthermore, we could determine the minimal sample size for a given scenario, which has to be collected to represent properly both geographic and feature space. In conclusion, SSA is a valuable tool to optimize the sampling design considering a lot of constraints.

A systematic comparison of summary characteristics for quantifying point patterns in ecology

Many functional summary characteristics such as Ripley's K function have been used in ecology to describe the spatial structure of point patterns to aid understanding of the underlying processes. However, their use is poorly guided in ecology because little is understood how well single summary characteristics, or a combination of them, capture the spatial structure of real world patterns. Here, we systematically tested the performance of combinations of eight summary characteristics [i.e. pair correlation function g(r), K‐function K(r), the proportion E(r) of points with no neighbor at distance r, the nearest neighbor distribution function D(r), the spherical contact distribution Hs(r), the kth nearest‐neighbor distribution functions Dk(r), the mean distance nn(k) to the kth neighbor, and the intensity function λ(x)]. To this end we used point pattern data covering a wide range of spatial structures including simulated (stationary) as well as real, possibly non‐stationary, patterns on tree species in a tropical forest in Panamá. To measure the information contained in a given combination of summary characteristics we used simulated annealing to reconstruct the observed patterns based only on the limited information provided by this combination and assessed how well other characteristics of the observed pattern were recovered.We found that the number of summary characteristics required to capture the spatial structure of stationary patterns varied between one (for patterns with near random structures) and three (for patterns with complex cluster and superposition structures), but with a robust ranking g(r), Dk(r), and Hs(r) that was largely independent on pattern idiosyncrasies. Stationary summary characteristics [with ranking g(r), Dk(r), Hs(r), E(r)] captured small‐ to intermediate scale properties of non‐stationary patterns, but for describing large‐scale spatial structures the intensity function was required. Our finding revealed that the current practice in ecology of using only one or two summary characteristics bears danger that essential characteristics of more complex patterns would not be detected. The technique of pattern reconstruction presented here has wide applications in ecology.

Structural properties of amorphous diamond-like carbon: percolation, cluster, and pair correlation analysis

A detailed atomistic model of amorphous diamond-like carbon was developed combining experimental neutron scattering data (K. W. R. Gilkes, P. H. Gaskell and J. Robertson, Phys. Rev. B, 1995, 51, 12303) with the hybrid reverse Monte Carlo method. From the experimentally consistent nanoscale model of the disordered tetrahedral carbon we computed various properties, including: binding energy distribution, neighbor distribution function, bond angle distribution, cluster size distributions for different C–C bond lengths, and percolation threshold. Analysis of microscopic configurations revealed that the network structure of the studied amorphous diamond-like carbon lacks any graphitic fragments (i.e., regular hexagons with a 120° C–C–C bond angle). We found that for the assumed C–C bond length ≤ 1.42 A (i.e., sp2 hybridization), the carbon network is poorly connected with a 70% contribution from isolated carbon atoms. The percolation threshold corresponds to a C–C bond length ≤ 1.52 A, which is close to 1.54 A (i.e., C–C bond length in perfect diamond). This finding is consistent with experimental high levels of tetrahedral bonding (i.e., sp3 hybridization) reported for high density tetrahedral amorphous carbons (i.e., sp3 fraction of 80–85%). Thus, we concluded that the HRMC method complemented with cluster size analysis and determination of percolation threshold is a promising methodology in studies of ill-defined carbonaceous materials.

Application of digital image processing for automatic detection and characterization of cracked constituent particles/inclusions in wrought aluminum alloys

Application of digital image processing is presented for automatic detection and segmentation of cracked iron-rich constituent particles in a hot-rolled wrought Al-alloy where the particle cracks are generated due to monotonic uniaxial tensile loading. Particle cracks (and therefore, cracked particles) with width of 0.5 µm or larger can be detected and segmented. The method is designed for detection of particle cracks generated during tensile testing, and not the particle cracks generated during the thermo-mechanical processing steps such as hot-rolling. The automatic segmentation of cracked iron-rich constituent particles enables robust estimation of second order microstructural descriptors such as nearest neighbor distribution and two-point correlation function of the cracked particles. Such data are very useful to verify and study the theoretical models for damage evolution and fracture in wrought Al-alloys.

Supramolecular photosystem II organization in grana thylakoid membranes: evidence for a structured arrangement.

The distribution of photosystem (PS) II complexes in stacked grana thylakoids derived from electron microscopic images of freeze-fractured chloroplasts are examined for the first time using mathematical methods. These characterize the particle distribution in terms of a nearest neighbor distribution function and a pair correlation function. The data were compared with purely random distributions calculated by a Monte Carlo simulation. The analysis reveals that the PSII distribution in grana thylakoids does not correspond to a random protein mixture but that ordering forces lead to a structured arrangement on a supramolecular level. Neighboring photosystems are significantly more separated than would be the case in a purely random distribution. These results are explained by structural models, in which boundary lipids and light-harvesting complex (LHC) II trimers are arranged between neighboring PSII. Furthermore, the diffusion of PSII was analyzed by a Monte Carlo simulation with a protein density of 80% area occupation (determined for grana membranes). The mobility of the photosystems is severely reduced by the high protein density. From an estimate of the mean migration time of PSII from grana thylakoids to stroma lamellae, it becomes evident that this diffusion contributes significantly to the velocity of the repair cycle of photoinhibited PSII.

Exact, numerical, and mean field behavior of a dimerizing lattice in one dimension

The thermodynamics and dynamics of a one-dimensional dimer-forming anharmonic model is studied in the classical limit. This model mimics the behavior of materials with a Peierls instability. Specific heat C( T), correlation length l c( T), and order parameter are calculated in three ways: (a) by mean-field (MF) approximation, (b) by numerical molecular dynamics (MD) simulation, and (c) by an exact transfer matrix method. The neighbor distribution function F( x) and vibrational density-of-states D( ω) (Fourier transform of velocity–velocity correlation function) are found numerically. MF theory fails completely to describe the destruction of long-range order by fluctuations, but nevertheless, the MF answers for C( T), l c( T), F( x), and D( ω) give helpful interpretations of the exact behavior.